Six letters J, K, L, M, N and O are written on different faces of a dice. Two positions of this dice are shown in the figure below. Find the letter on the face opposite to J.

Correct — A. Read both pictures first. View 1 shows M on top, N in front and O on the right; view 2 shows K on top, J in front and O on the right.

Rule: a face that stays in the same position in both views fixes the axis of turn, and the four faces around it form a ring.

O is on the right in both, so M, N, K and J all touch O. That is the full ring of four, which leaves L opposite O.

How far did it turn? A quarter turn about that axis would bring the old front onto the top (N) or the old top to the front (M). Neither happens.

So it is a half turn: top swaps with bottom, front swaps with back, right stays right. The new top K was the old bottom, and the new front J was the old back.

That makes K opposite M and J opposite N — option (a).

Why the others are wrong

• (b)
  M — M is opposite K, not J. Under the half turn the old top M went to the bottom, and view 2 shows K standing in its place on top.
• (c)
  O — O holds the right-hand face in both views, so it is never on the top or the front. Its partner is the one letter no picture shows.
• (d)
  L — L is hidden in both views, and that is exactly what marks it as the face opposite O — the only letter neither picture can reach.

A two-view dice question with a common face, which is the friendliest form it takes.

When one face keeps the same position in both views, the die has been turned about the axis through that face. Everything else you need follows from two facts: the four faces around that axis form a ring, and any face you can see in the same picture as another is adjacent to it, never opposite.

Counting the ring is the first move. Four distinct letters appear beside O across the two views, so the ring is complete and the sixth letter must be O's own partner.

Key facts

• View 1 shows M on top, N in front and O on the right.
• View 2 shows K on top, J in front and O on the right.
• M, N, K and J are all adjacent to O, so L is opposite O.
• A half turn about the left-right axis swaps top with bottom and front with back, giving M opposite K and N opposite J.

Common traps

• Assuming any quarter turn without checking it against the letters actually shown.
• Treating two faces visible in the same picture as opposite.
• Forgetting that the letter absent from both views is the partner of the face that never moved.

No directly related past PYQ was found.